Theorem scafvalg 14184

Description: The scalar multiplication operation as a function. (Contributed by Mario Carneiro, 5-Oct-2015.)

Hypotheses

Ref	Expression
scaffval.b	|- B = ( Base ` W )
scaffval.f	|- F = (Scalar ` W )
scaffval.k	|- K = ( Base ` F )
scaffval.a	|- .xb = ( .sf ` W )
scaffval.s	|- .x. = ( .s ` W )

Assertion

Ref	Expression
scafvalg	|- ( ( W e. V /\ X e. K /\ Y e. B ) -> ( X .xb Y ) = ( X .x. Y ) )

Proof of Theorem scafvalg

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		scaffval.b	. . . 4 |- B = ( Base ` W )
2		scaffval.f	. . . 4 |- F = (Scalar ` W )
3		scaffval.k	. . . 4 |- K = ( Base ` F )
4		scaffval.a	. . . 4 |- .xb = ( .sf ` W )
5		scaffval.s	. . . 4 |- .x. = ( .s ` W )
6	1, 2, 3, 4, 5	scaffvalg 14183	. . 3 |- ( W e. V -> .xb = ( x e. K , y e. B |-> ( x .x. y ) ) )
7	6	3ad2ant1 1021	. 2 |- ( ( W e. V /\ X e. K /\ Y e. B ) -> .xb = ( x e. K , y e. B |-> ( x .x. y ) ) )
8		oveq12 5976	. . 3 |- ( ( x = X /\ y = Y ) -> ( x .x. y ) = ( X .x. Y ) )
9	8	adantl 277	. 2 |- ( ( ( W e. V /\ X e. K /\ Y e. B ) /\ ( x = X /\ y = Y ) ) -> ( x .x. y ) = ( X .x. Y ) )
10		simp2 1001	. 2 |- ( ( W e. V /\ X e. K /\ Y e. B ) -> X e. K )
11		simp3 1002	. 2 |- ( ( W e. V /\ X e. K /\ Y e. B ) -> Y e. B )
12		vscaslid 13110	. . . . . 6 |- ( .s = Slot ( .s ` ndx ) /\ ( .s ` ndx ) e. NN )
13	12	slotex 12974	. . . . 5 |- ( W e. V -> ( .s ` W ) e. _V )
14	5, 13	eqeltrid 2294	. . . 4 |- ( W e. V -> .x. e. _V )
15	14	3ad2ant1 1021	. . 3 |- ( ( W e. V /\ X e. K /\ Y e. B ) -> .x. e. _V )
16		ovexg 6001	. . 3 |- ( ( X e. K /\ .x. e. _V /\ Y e. B ) -> ( X .x. Y ) e. _V )
17	10, 15, 11, 16	syl3anc 1250	. 2 |- ( ( W e. V /\ X e. K /\ Y e. B ) -> ( X .x. Y ) e. _V )
18	7, 9, 10, 11, 17	ovmpod 6096	1 |- ( ( W e. V /\ X e. K /\ Y e. B ) -> ( X .xb Y ) = ( X .x. Y ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 981   = wceq 1373   e. wcel 2178   _Vcvv 2776   ` cfv 5290  (class class class)co 5967   e. cmpo 5969   Basecbs 12947  Scalarcsca 13027   .scvsca 13028   .sfcscaf 14165

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1re 8054  ax-addrcl 8057

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-inn 9072  df-2 9130  df-3 9131  df-4 9132  df-5 9133  df-6 9134  df-ndx 12950  df-slot 12951  df-base 12953  df-sca 13040  df-vsca 13041  df-scaf 14167

This theorem is referenced by:  lmodfopne  14203